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The Parisi Formula for the Edwards–Anderson Model in the High-Dimensional Limit

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.MP — Mathematical Physics

Contributed by P. M. Aronow ↗claimed, Patrick Lopatto ↗

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Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

We study the Edwards–Anderson spin glass on the torus (Z/LZ)d(\mathbb{Z}/L\mathbb{Z})^d, with independent Gaussian couplings of variance 1/(2d)1/(2d) between nearest neighbors. We prove that, as d→∞d\to\infty, its free energy converges to that of the Sherrington–Kirkpatrick model, given by the Parisi formula, at every temperature and in every uniform external field, uniformly in the side length LL. The same holds on the hypercube {0,1}d\{0,1\}^d, and the ground-state energies converge to the Sherrington–Kirkpatrick ground-state energy.

Provenance statement

Proof discovery and verification were done with a combination of GPT 6 Astra, GPT 5.6 Sol, and GPT 6.1 Sol. The writing was done by Opus 5.5, with final revisions from GPT 6 Astra.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion 5.6
OpenAI
ChatGPTVersion 6
OpenAI
ChatGPTVersion 6.1

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Version history

  1. v1Submitted by Patrick LopattoInitial depositCurrentOct 08, 2026